To fully specify the estimate of \(g_j\) , some hyperparameters need to be determined. This includes the regularization parameter, \(\lambda \)  (related to the cost parameter in a typical SVM), the radius of the tube, \(\epsilon \) , for the loss function, and some kernel related parameters. For this hyperparameter learning, we apply grid search based on generalized cross-validation (GCV) criterion. The proposed method already employs 5-fold CV for the selection of critical subspaces and the termination of the iterative search, so another layer of out-of-sample prediction will complicate the data structure and the training dataset may not suffer from the lack of a sufficient number of data points. While measuring in-sample error using the training data only, GCV provides a convenient approximation to leave-one-out CV [1]}, so it is a proper criterion for the hyperparameter learning of the proposed method. In general, GCV is calculated as
\(GCV(\hat{f}) = \frac{1}{n}\sum _{i=1}^{n} \Big [\frac{y_i- \hat{f}(\mathbf {x}_i)}{1-\text{trace}(\mathbf {S})/n)}\Big ]^2\) 
